General Cheeger inequalities for p-Laplacians on graphs

TL;DR

This paper establishes new Cheeger inequalities for p-Laplacians on graphs without bounded degree restrictions, using intrinsic metrics to improve bounds on spectral gaps for both finite and infinite graphs.

Contribution

It introduces a novel boundary measure based on intrinsic metrics, enabling Cheeger inequalities without degree boundedness assumptions.

Findings

Improved bounds on spectral gaps for finite graphs.

Non-trivial bounds for infinite graphs with unbounded degrees.

Upper bounds via Cheeger constant and volume growth.

Abstract

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which is using the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls.